# examples of probable primes

To give an example of a probable prime relative to a base: $4^{341233}-3^{341233}$ has passed preliminary primality tests relative to bases 2, 3, 5, 7, 11, 13 and 101. Its square root is approximately $2.3362\cdot 10^{102721}$, which makes a conclusive primality test by trial division in a reasonable time period impractical.

To give an example of a probable prime by a pattern: this pattern

 $2^{2}-1=3,2^{3}-1=7,2^{7}-1=127$
 $2^{127}-1=170141183460469231731687303715884105727$

suggests that $2^{170141183460469231731687303715884105727}-1$ might be a Mersenne prime. But since this is larger than the largest known Mersenne prime $2^{30402457}-1$ (as of 2005), a Lucas-Lehmer test might take longer than the average human lifetime.

On the other hand, $123456789\cdot 10^{123456789}+123456789$ is not a probable prime, because even though it is much larger than either of the probable primes given above, it is clearly divisible by $3^{2}$.

Title examples of probable primes ExamplesOfProbablePrimes 2013-03-22 15:53:49 2013-03-22 15:53:49 PrimeFan (13766) PrimeFan (13766) 5 PrimeFan (13766) Example msc 11A41